CHAPTER 12 Hedging hedging dddddddddddddd ddd hedging strategy = replicating strategy hedgingdd) ddd Question : How to find a hedging strategy? In other words, for an attainable contingent claim, find a self-financing strategy η, ξ) such that C = η t B t + ξ t S t 121 Hedging strategy for simple contingent claim form Consider the generalized Black-Scholes model and a simple contingent claim C of the C =ΦS T ) Assumption : The price process πt, Φ) is given by π t = πt, Φ) = F t, S t ) 283
284 12 HEDGING dd: dddd price process ddddddd) By Itô s formula dπ t = df t, S t ) = F t t, S t ) dt + F x t, S t ) ds t + 1 2 F xxt, S t )ds t ) 2 = F t dt + F x S t αdt+ S t σdw t )+ 1 2 F xxst 2 σ 2 dt = F t + αs t F x + 12 ) σ2 S 2t F xx dt + σs t F x dw t = α π π t dt + σ π π t dw t, 121) where α π = F t + αs t F x + 1 2 σ2 St 2 F xx, F σ π = σs tf x F Goal : Find a self-financing strategy η t,ξ t ) such that V t = η t B t + ξ t S t = π t = πt, S t ) for all t Idea :IfV t = π t, by 121), we have dv t = dπ t = α π π t dt + σ π π t dw t Moreover, since η t,ξ t ) is self-financing, dv t = η t db t + ξ t ds t = rη t B t dt + ξ t S t αdt+ σdw t )
This implies that 121 HEDGING STRATEGY FOR SIMPLE CONTINGENT CLAIM 285 σ π π t = σs t ξ t, α π π t = rη t B t + αξ t S t However, since F = V t = π t, and Thus, ξ t = σ ππ t σs t = F x, η t = α ππ t αξ t S t rb t = F t + 1 2 σ2 S 2 t F xx rb t F t + 1 2 σ2 S 2 t F xx rb t = η t = F S tf x B t This implies that F t + rs t F x + 1 2 σ2 S 2 t F xx rf = Remark 121 The following contingent claims satisfy the above assumptions: European call option C =S T K) + forward contract C = S T K However, not every contingent satisfies the assumption, eg, Asian option C = 1 T T ) + S t dt K lookback contract C = S T inf t T S t
286 12 HEDGING 122 Delta and gamma hedging Consider a financial market Definition 122 A portfolio which is insensitive with respect to small changes in one of the greeks is said to be neutral In particular, 1) a portfolio with zero delta is said to be delta neutral 2) a portfolio with zero gamma is said to be gamma neutral ddddddddd portfolio dddddd, dddd portfolio dddddddd, d ddd the valve of the portfolio Consider a contingent claim with pricing function F t, S t ) Goal : Immunizing this portfolio against small changes in the underlying asset price S t Ft, x) If the portfolio is delta neutral, ie, Δ F = =, done! x If not, sell the entire portfolio But not feasible, preferable! An interesting idea : add a derivative eg, an option or the underlying asset itself) to the portfolio Denote the pricing function of the chosen derivative by P t, S t ) and consider a portfolio 1,ξ), ie, own one unit of the first asset and ξ unit of the additional derivative Then the value of the portfolio is given by V t, S t )=F t, S t )+ξpt, S t ) In order to make this portfolio delta neutral, we have to find ξ such that V x =
122 DELTA AND GAMMA HEDGING 287 This implies that F x + ξ P x =, ie, ΔF + ξδp = Thus, ξ is given by ξ = ΔF ΔP Remark 123 If the additional derivative is the undrlying asset, then the stock price at time t is P t, x) =πt, Φ s )=x, and ΔP =1 This implies that ξ = ΔF, and the delta of a derivative give us the number of units of the underlying stock that is needed in order to hedge the derivative dddddd: ddd stratgey, dddddddddd dddddd, ddddd ddddd, ddddddddddddd, dddd discrete rebalanced delta hedge ddddddd rebalance? If we rebalance often, it is a very good hedge, but suffer from high transaction costs dddd, dddddddddddddd ddddddddd? ddddddddddddddddddddd If the gamma Γ = Δ x = 2 F x 2 is high, we have to rebalance often Hence, a low gamma will allow us to keep the delta hedge for a long period dddd delta neutral dd, ddd dd gamma neutral d gamma neutral dddddd delta neutral dddddd Fixed two derivatives with pricing functions P t, S t ) and Qt, S t ) Consider a portfolio 1,η t,ξ t ) Then the value of the portfolio is given by V t, S t )=F t, S t )+η t P t, S t )+ξ t Qt, S t )
288 12 HEDGING Due to delta neutral, and gamma neutral, V =, ie, x 2 V =, ie, x2 Δ F + η t Δ P + ξ t Δ Q = Γ F + η t Γ P + ξ t Γ Q =, we can solve the system of linear equations to get η t and ξ t Hence Δ F Δ Q Γ F Γ Q η t = = Δ QΓ F Δ F Γ Q, Δ P Δ Q Δ P Γ Q Δ Q Γ P Γ P Γ Q Δ P Δ F Γ P Γ F ξ t = = Δ F Γ P Δ P Γ F Δ P Δ Q Δ P Γ Q Δ Q Γ P Consider a special case Γ P Γ Q Lemma 124 For the underlying stock, Δ S =1, Γ S = Thus, we have the following result Remark 125 If P t, S t )=S t, then η t = Δ QΓ F Δ F Γ Q Γ Q = Δ Q Γ Q Γ F Δ F, ξ t = Γ F Γ Q
123 SUPERHEDGING 289 123 Superhedging ddddddddd complete market case Explicitly, dddddd hedgable / attainable contingent claim d hedging strategies dd incomplete market ddddd non-attainable contingent claim d hedging ddddd Definition 126 Let H be a contingent claim 1) If H is attainable, the smallest initial wealth allowing to reach H at maturity is an attainable strategy is called hedging price of H 2) A super-hedging strategy for H is an admissible trading strategy such that V T H 3) The super-hedging price of H is the smallest initial endowment of a super-hedging strategy ddddd hedging price ddd Example 127 In a standard Black-Scholes model, consider the European call option C =S T K) + By Section 121, the hedging strategy is given by ξ t = F x t, S t ), η t = F S tf x B t Since the option price at time t is given by F t, S t )=S t N d 1 ) Ke rt t) N d 2 ),
29 12 HEDGING where d 1 = d 2 = 1 σ T t 1 σ T t ln ln St ) + r + σ2 K 2 ) + r σ2 2 St K ) ) T t), ) ) T t) and F x t, S t ) = N d 1 ), F t t, S t ) = σs t 2 d 2 2πT t) e 1 2 rke rt t) N d 2 ), the hedging strategy is given by ξ t = N d 1 ), η t = e rt [S t N d 1 ) Ke rt t) N d 2 ) S t N d 1 )] = Ke rt N d 2 ) In particular, if t = d 1 = d 2 = 1 σ T 1 σ T ln ln S ) + r + σ2 K 2 ) + r σ2 2 S K ) ) T, ) ) T, the hedging price is of the form F,S )=S N d 1 ) Ke rt N d 2 ), which is coincide with the option price at time ddd, Super-hedging price d contingent claim d price ddddddd
123 SUPERHEDGING 291 Notation 128 A = the collection of all admissible strategies, ie, A is the collection of all pairs x, θ), where x is the initial endowment and θ is a predictable process satisfying Ṽ t x, θ) =x + for all t [,T] and for some A> t θ u d S u A, Definition 129 1) x, θ) Ais a super-hedging strategy for H if Ṽ x,θ T H B T, ie, V T x, θ) H 2) The super-hedging price ΠH) of H is given by πh) = inf{x R, x, θ) A,V T x, θ) H as} Theorem 121 Super-hedging theorem) Let P be the collection of all equivalent local martingale measures Then the super-hedging price πh) satisfies [ ] H πh) = sup E Q Q P B T In general, the wealth dynamics of the minimal super-hedging portfolio for H is given by V t = ess sup Q P B t E Q [ H B T F t ] ddddddddddddd, dd, d superhedging price dddddddddd ddddd dddddddddddddd quantile hedging dd super-hedging d dd, ddddddddd Luciano Campi [5] d N E1 Karoui and M-C Quenez [14] dd superhedging strategy ddddd DOKramkov [26]
292 12 HEDGING 124 Quantile hedging Superhedging dddddddd, dddd perfect hedging dddddd initial endowment, dddddd, dddddddddd quantile hedging Quantile hedging ddddddd, dddddd 1% perfect hedging d, hedging price dddddd Reference dd H Föllmer and P Leukert [16] ddddddd complete market case ie, the contingent claim H is attainable and the interest rate r =, ie, there exists a self-financing strategy ξ H such that H = H B T = V + T ξ H u dtildes u Moreover, the discounted value process satisfies [ ] H E [H F t ]=E F t = B Ṽt = V + T This implies that the hedging price t ξ H u d S u H = E [ H B T ] = E [H] Question : What is the best hedge the investor can achieve with a given smaller amount V <H? Thus, we ate looking for an admissible strategy V,ξ) such that with constraint V V [ T ]! P V + ξ u ds u H = max 122) Definition 1211 The set {V T H} is called the success set corresponding to the admissible strategy V,ξ)
124 QUANTILE HEDGING 293 Proposition 1212 Let à F i be the solution of the problem PA)! = max where A = {V T H}) under the constraint E [HI A ] V Let ξ denote the perfect hedge for the knockout option tildeh = HIà L 1 P ) Then V, ξ) solves the optimalization problem 122), and the coresponding success set = à as An alternative question : Given α, 1) Find V := inf { X : admissible strategy ξ such that T ) } P X + ξ u ds u H 1 α Example 1213 Consider the standard Black-Scholes model with σ = 3, μ = 8, S = 1, K = 11 Then α 1 5 1 V /H 89 59 34